APPLIED MATH, PAPER-I
FEDERAL PUBLIC SERVICE COMMISSION
COMPETITIVE EXAMINATION FOR
RECRUITMENT TO POSTS IN BPS-17 UNDER
THE FEDERAL GOVERNMENT, 2009
APPLIED MATH, PAPER-I
TIME ALLOWED: 3 HOURS MAXIMUM MARKS:100
NOTE:
(i) Attempt FIVE question in all by selecting at least TWO questions from SECTION – A
and THREE question from SECTION – B. All questions carry EQUAL marks.
(ii) Use of Scientific Calculator is allowed.
SECTION – A
Q.1. (a) Show that in orthogonal coordinates: (5+5)
(i) ( ) ( ) ( 1 1 ),
1 2 2
3
1 1
1 3 3
2
1 1 A h
h h u
e
A h
h h u
e
A e
∂
∂
−
∂
∂
∇ × =
(ii) ( ) 1 ( ).
1 2 3
1 2 3 1
1 1 A h h
h h h u
A e
∂
∂
∇ • =
(b) Write Laplace ’s equation in parabolic cylindrical coordinates. (10)
Q.2. (a) Evaluate ∫∫ •
S
A nds
ρ ρ
, where A = ziˆ + xˆj −3y 2 kˆ
ρ
and s is the surface of cylinder x2+y2=16
included in the first octant between z=0 and z=5. (10)
(b) Verify Green’s theorem in the plane for
∫ ( + ) +
c
xy y 2 dx x2dy
where C is the closed curve of the region bounded by y = x and y = x2.
Q.3. (a) Find centre of mass of a right circular solid cone of height h. (10)
(b) A light thin rod, 4m long, can turn in a vertical plane about one of its point which is
attached to a pivot. If weights of 3kg and 4kg are suspended from its ends it rests in a
horizontal position. Find the position of the pivot and its reaction on the rod. (10)
SECTION – B
Q.4. (a) Find the radial and transverse components of the velocity of a particle moving along
the curve ax2+by2=1 at any time t if the polar angle θ = ct2. (10)
(b) A particle is projected vertically upwards. After a time t, another particle is sent up
from the same point with the same velocity and meets the first a height h during the
downward flight of the first. Find the velocity of projection. (10)
Q.5. (a) If a point P moves with a velocity v given by
v2=n2 (ax2+bx+c),
show that P executes a simple harmonic motion. Find also, the centre, the amplitude
and the time period of the motion. (10)
S.No.
R.No.
APPLIED MATH, PAPER-I
(b) A particle of mass m moves on xy-plane under the force (10)
, 4 r
r
F k ϖ ϖ
= −
where r is its distance from the origin O. If it starts on the positive x-axis at a distance “a” from
O with speed vo in a direction making an angle θ with the positive x-direction, prove that at
time t,
3
2 2 sin 2
mr
ma v k
r o −
=
θ
&&
Q.6. (a) Define angular momentum and prove that rate of change of angular momentum of a
particle about a point O is equal to the tarque (about O) of the force acting on the particle. (10)
(b) Find the least speed with which a particle must be projected so that it passes through two
points P and Q at heights h1 and h2, respectively. (10)
Q.7. (a) Discuss the polar form of an orbit and prove that when a particle moves under central
force, the areal velocity is constant. (10)
(b) A particle moves under a central repulsive force μ/r3 and is projected from an apse at a
distance “a” with velocity V. Show that the equation to the path is (10)
r cosp θ = a
and that the angle θ described in time t is
1 tan 1 ,
a
pVt
p
−
where 1 , . 2 2
2 GM
a v
p = + μ =
μ
Q.8. (a) Define the terms moment of inertia and product of inertia, and find the moment of
inertia of uniform solid sphere of mass m and radius “a”. (10)
(b) Let AB and BC be two equal similar rods freely hinged at B and lie in a straight line
on the smooth table. The end A is struck by a blow P perpendicular to AB. Show
that the resulting velocity of A is
2
3 1 times that of B. (10)
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